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Theorem pnfnre 8029
Description: Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
pnfnre  |- +oo  e/  RR

Proof of Theorem pnfnre
StepHypRef Expression
1 cnex 7965 . . . . . 6  |-  CC  e.  _V
21uniex 4455 . . . . 5  |-  U. CC  e.  _V
3 pwuninel2 6307 . . . . 5  |-  ( U. CC  e.  _V  ->  -.  ~P U. CC  e.  CC )
42, 3ax-mp 5 . . . 4  |-  -.  ~P U. CC  e.  CC
5 df-pnf 8024 . . . . 5  |- +oo  =  ~P U. CC
65eleq1i 2255 . . . 4  |-  ( +oo  e.  CC  <->  ~P U. CC  e.  CC )
74, 6mtbir 672 . . 3  |-  -. +oo  e.  CC
8 recn 7974 . . 3  |-  ( +oo  e.  RR  -> +oo  e.  CC )
97, 8mto 663 . 2  |-  -. +oo  e.  RR
109nelir 2458 1  |- +oo  e/  RR
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 2160    e/ wnel 2455   _Vcvv 2752   ~Pcpw 3590   U.cuni 3824   CCcc 7839   RRcr 7840   +oocpnf 8019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-un 4451  ax-cnex 7932  ax-resscn 7933
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-nel 2456  df-rex 2474  df-rab 2477  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-pnf 8024
This theorem is referenced by:  renepnf  8035  nn0nepnf  9277  xrltnr  9809  pnfnlt  9817  xnn0lenn0nn0  9895  inftonninf  10472  pcgcd1  12360  pc2dvds  12362
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